HDU-4336 Card Collector 概率DP

　　题目链接：http://acm.hdu.edu.cn/showproblem.php?pid=4336

　　题意：买食品收集n个卡片，每个卡片的概率分别是pi，且Σp[i]<=1，求收集n个卡片需要买的食品数的期望。

　　压缩DP：把每个食品用二进制表示，0和1分别表示没有卡片和已经收集到此卡片的期望，则

　　　　 f[s]=(1-Σp[i])*f[s]+Σp[j]*f[s]+Σp[k]*f[s|(1<<k)]  

　　　　　　s表示状态，i表示所有卡片编号，j表示s状态中已经有的卡片编号，k表示s状态中没有的卡片编号

　　->  Σp[i]*f[s]=Σp[i]*f[s|(1<<i)] 

　　或者容斥原理做：

　　压缩DP:

 1 //STATUS:C++_AC_281MS_7128KB

 2 #include <functional>

 3 #include <algorithm>

 4 #include <iostream>

 5 //#include <ext/rope>

 6 #include <fstream>

 7 #include <sstream>

 8 #include <iomanip>

 9 #include <numeric>

10 #include <cstring>

11 #include <cassert>

12 #include <cstdio>

13 #include <string>

14 #include <vector>

15 #include <bitset>

16 #include <queue>

17 #include <stack>

18 #include <cmath>

19 #include <ctime>

20 #include <list>

21 #include <set>

22 #include <map>

23 using namespace std;

24 //#pragma comment(linker,"/STACK:102400000,102400000")

25 //using namespace __gnu_cxx;

26 //define

27 #define pii pair<int,int>

28 #define mem(a,b) memset(a,b,sizeof(a))

29 #define lson l,mid,rt<<1

30 #define rson mid+1,r,rt<<1|1

31 #define PI acos(-1.0)

32 //typedef

33 typedef __int64 LL;

34 typedef unsigned __int64 ULL;

35 //const

36 const int N=(1<<20)+10;

37 const int INF=0x3f3f3f3f;

38 const int MOD= 1000000007,STA=8000010;

39 const LL LNF=1LL<<55;

40 const double EPS=1e-9;

41 const double OO=1e30;

42 const int dx[4]={-1,0,1,0};

43 const int dy[4]={0,1,0,-1};

44 const int day[13]={0,31,28,31,30,31,30,31,31,30,31,30,31};

45 //Daily Use ...

46 inline int sign(double x){return (x>EPS)-(x<-EPS);}

47 template<class T> T gcd(T a,T b){return b?gcd(b,a%b):a;}

48 template<class T> T lcm(T a,T b){return a/gcd(a,b)*b;}

49 template<class T> inline T lcm(T a,T b,T d){return a/d*b;}

50 template<class T> inline T Min(T a,T b){return a<b?a:b;}

51 template<class T> inline T Max(T a,T b){return a>b?a:b;}

52 template<class T> inline T Min(T a,T b,T c){return min(min(a, b),c);}

53 template<class T> inline T Max(T a,T b,T c){return max(max(a, b),c);}

54 template<class T> inline T Min(T a,T b,T c,T d){return min(min(a, b),min(c,d));}

55 template<class T> inline T Max(T a,T b,T c,T d){return max(max(a, b),max(c,d));}

56 //End

57 

58 double p[23],f[N];

59 int n;

60 

61 int main(){

62  //   freopen("in.txt","r",stdin);

63     int i,j,up;

64     double s;

65     while(~scanf("%d",&n))

66     {

67         for(i=0;i<n;i++){

68             scanf("%lf",&p[i]);

69         }

70         up=(1<<n)-1;

71         f[up]=0;

72         for(i=up-1;i>=0;i--){

73             f[i]=1;s=0;

74             for(j=0;j<n;j++){

75                 if(i&(1<<j))continue;

76                 f[i]+=p[j]*f[i|(1<<j)];

77                 s+=p[j];

78             }

79             f[i]/=s;

80         }

81 

82         printf("%lf\n",f[0]);

83     }

84     return 0;

85 }

 

　　容斥原理：

 1 //STATUS:C++_AC_203MS_244KB

 2 #include <functional>

 3 #include <algorithm>

 4 #include <iostream>

 5 //#include <ext/rope>

 6 #include <fstream>

 7 #include <sstream>

 8 #include <iomanip>

 9 #include <numeric>

10 #include <cstring>

11 #include <cassert>

12 #include <cstdio>

13 #include <string>

14 #include <vector>

15 #include <bitset>

16 #include <queue>

17 #include <stack>

18 #include <cmath>

19 #include <ctime>

20 #include <list>

21 #include <set>

22 #include <map>

23 using namespace std;

24 //#pragma comment(linker,"/STACK:102400000,102400000")

25 //using namespace __gnu_cxx;

26 //define

27 #define pii pair<int,int>

28 #define mem(a,b) memset(a,b,sizeof(a))

29 #define lson l,mid,rt<<1

30 #define rson mid+1,r,rt<<1|1

31 #define PI acos(-1.0)

32 //typedef

33 typedef __int64 LL;

34 typedef unsigned __int64 ULL;

35 //const

36 const int N=(1<<20)+10;

37 const int INF=0x3f3f3f3f;

38 const int MOD= 1000000007,STA=8000010;

39 const LL LNF=1LL<<55;

40 const double EPS=1e-9;

41 const double OO=1e30;

42 const int dx[4]={-1,0,1,0};

43 const int dy[4]={0,1,0,-1};

44 const int day[13]={0,31,28,31,30,31,30,31,31,30,31,30,31};

45 //Daily Use ...

46 inline int sign(double x){return (x>EPS)-(x<-EPS);}

47 template<class T> T gcd(T a,T b){return b?gcd(b,a%b):a;}

48 template<class T> T lcm(T a,T b){return a/gcd(a,b)*b;}

49 template<class T> inline T lcm(T a,T b,T d){return a/d*b;}

50 template<class T> inline T Min(T a,T b){return a<b?a:b;}

51 template<class T> inline T Max(T a,T b){return a>b?a:b;}

52 template<class T> inline T Min(T a,T b,T c){return min(min(a, b),c);}

53 template<class T> inline T Max(T a,T b,T c){return max(max(a, b),c);}

54 template<class T> inline T Min(T a,T b,T c,T d){return min(min(a, b),min(c,d));}

55 template<class T> inline T Max(T a,T b,T c,T d){return max(max(a, b),max(c,d));}

56 //End

57 

58 double p[23];

59 int n;

60 

61 int main(){

62  //   freopen("in.txt","r",stdin);

63     int i,j,up,cnt;

64     double ans,s;

65     while(~scanf("%d",&n))

66     {

67         for(i=0;i<n;i++){

68             scanf("%lf",&p[i]);

69         }

70         up=(1<<n)-1;ans=0;

71         for(i=1;i<=up;i++){

72             s=0;

73             for(j=cnt=0;j<n;j++){

74                 if(i&(1<<j)){

75                     cnt++;

76                     s+=p[j];

77                 }

78             }

79             if(cnt&1)ans+=1/s;

80             else ans-=1/s;

81         }

82 

83         printf("%lf\n",ans);

84     }

85     return 0;

86 }